# 12.4 Integrated rate laws  (Page 5/7)

 Page 5 / 7

The half-life of a reaction is the time required to decrease the amount of a given reactant by one-half. The half-life of a zero-order reaction decreases as the initial concentration of the reactant in the reaction decreases. The half-life of a first-order reaction is independent of concentration, and the half-life of a second-order reaction decreases as the concentration increases.

## Key equations

• integrated rate law for zero-order reactions: $\left[A\right]=\text{−}kt+{\left[A\right]}_{0},$ ${t}_{1\text{/}2}=\phantom{\rule{0.1em}{0ex}}\frac{{\left[A\right]}_{0}}{2k}$
• integrated rate law for first-order reactions: $\text{ln}\left[A\right]=\text{−}kt+\text{ln}{\left[A\right]}_{0},\phantom{\rule{0.2em}{0ex}}\text{}{t}_{1\text{/}2}=\phantom{\rule{0.1em}{0ex}}\frac{0.693}{k}$
• integrated rate law for second-order reactions: $\frac{1}{\left[A\right]}\phantom{\rule{0.1em}{0ex}}=kt+\phantom{\rule{0.2em}{0ex}}\frac{1}{{\left[A\right]}_{0}},$ ${t}_{1\text{/}2}=\phantom{\rule{0.1em}{0ex}}\frac{1}{{\left[A\right]}_{0}k}$

## Chemistry end of chapter exercises

Describe how graphical methods can be used to determine the order of a reaction and its rate constant from a series of data that includes the concentration of A at varying times.

Use the data provided to graphically determine the order and rate constant of the following reaction: ${\text{SO}}_{2}{\text{Cl}}_{2}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\phantom{\rule{0.2em}{0ex}}\text{SO}}_{2}+{\text{Cl}}_{2}$

 Time (s) 0 5.00 $×$ 10 3 1.00 $×$ 10 4 1.50 $×$ 10 4 [SO 2 Cl 2 ] ( M ) 0.100 0.0896 0.0802 0.0719 Time (s) 2.50 $×$ 10 4 3.00 $×$ 10 4 4.00 $×$ 10 4 [SO 2 Cl 2 ] ( M ) 0.0577 0.0517 0.0415

Plotting a graph of ln[SO 2 Cl 2 ] versus t reveals a linear trend; therefore we know this is a first-order reaction:

k = −2.20 $×$ 10 5 s −1

Use the data provided in a graphical method to determine the order and rate constant of the following reaction:
$2P\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}Q+W$

 Time (s) 9.0 13.0 18.0 22.0 25.0 [P] ( M ) 1.077 $×$ 10 −3 1.068 $×$ 10 −3 1.055 $×$ 10 −3 1.046 $×$ 10 −3 1.039 $×$ 10 −3

Pure ozone decomposes slowly to oxygen, ${\text{2O}}_{3}\left(g\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{3O}}_{2}\left(g\right).$ Use the data provided in a graphical method and determine the order and rate constant of the reaction.

 Time (h) 0 2.0 $×$ 10 3 7.6 $×$ 10 3 1.00 $×$ 10 4 [O 3 ] ( M ) 1.00 $×$ 10 −5 4.98 $×$ 10 −6 2.07 $×$ 10 −6 1.66 $×$ 10 −6 Time (h) 1.23 $×$ 10 4 1.43 $×$ 10 4 1.70 $×$ 10 4 [O 3 ] ( M ) 1.39 $×$ 10 −6 1.22 $×$ 10 −6 1.05 $×$ 10 −6

The plot is nicely linear, so the reaction is second order.
k = 50.1 L mol −1 h −1

From the given data, use a graphical method to determine the order and rate constant of the following reaction:
$2X\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}Y+Z$

 Time (s) 5 10 15 20 25 30 35 40 [ X ] ( M ) 0.099 0.0497 0.0332 0.0249 0.02 0.0166 0.0143 0.0125

What is the half-life for the first-order decay of phosphorus-32? $\left({}_{15}^{32}\text{P}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{}_{16}^{32}\text{S}+{\text{e}}^{-}\right)$ The rate constant for the decay is 4.85 $×$ 10 −2 day −1 .

14.3 d

What is the half-life for the first-order decay of carbon-14? $\left({}_{\phantom{\rule{0.5em}{0ex}}6}^{14}\text{C}⟶{}_{\phantom{\rule{0.5em}{0ex}}7}^{14}\text{N}+{\text{e}}^{-}\right)$ The rate constant for the decay is 1.21 $×$ 10 −4 year −1 .

What is the half-life for the decomposition of NOCl when the concentration of NOCl is 0.15 M ? The rate constant for this second-order reaction is 8.0 $×$ 10 −8 L/mol/s.

8.3 $×$ 10 7 s

What is the half-life for the decomposition of O 3 when the concentration of O 3 is 2.35 $×$ 10 −6 M ? The rate constant for this second-order reaction is 50.4 L/mol/h.

The reaction of compound   A to give compounds   C and   D was found to be second-order in   A . The rate constant for the reaction was determined to be 2.42 L/mol/s. If the initial concentration is 0.500 mol/L, what is the value of t 1/2 ?

0.826 s

The half-life of a reaction of compound A to give compounds D and E is 8.50 min when the initial concentration of A is 0.150 mol/L. How long will it take for the concentration to drop to 0.0300 mol/L if the reaction is (a) first order with respect to A or (b) second order with respect to A ?

Some bacteria are resistant to the antibiotic penicillin because they produce penicillinase, an enzyme with a molecular weight of 3 $×$ 10 4 g/mol that converts penicillin into inactive molecules. Although the kinetics of enzyme-catalyzed reactions can be complex, at low concentrations this reaction can be described by a rate equation that is first order in the catalyst (penicillinase) and that also involves the concentration of penicillin. From the following data: 1.0 L of a solution containing 0.15 µg (0.15 $×$ 10 −6 g) of penicillinase, determine the order of the reaction with respect to penicillin and the value of the rate constant.

[Penicillin] ( M ) Rate (mol/L/min)
2.0 $×$ 10 −6 1.0 $×$ 10 −10
3.0 $×$ 10 −6 1.5 $×$ 10 −10
4.0 $×$ 10 −6 2.0 $×$ 10 −10

The reaction is first order.
k = 1.0 $×$ 10 7 mol −1 min −1

Both technetium-99 and thallium-201 are used to image heart muscle in patients with suspected heart problems. The half-lives are 6 h and 73 h, respectively. What percent of the radioactivity would remain for each of the isotopes after 2 days (48 h)?

There are two molecules with the formula C 3 H 6 . Propene, ${\text{CH}}_{3}\text{CH}={\text{CH}}_{2},$ is the monomer of the polymer polypropylene, which is used for indoor-outdoor carpets. Cyclopropane is used as an anesthetic:

When heated to 499 °C, cyclopropane rearranges (isomerizes) and forms propene with a rate constant of
5.95 $×$ 10 −4 s −1 . What is the half-life of this reaction? What fraction of the cyclopropane remains after 0.75 h at 499.5 °C?

4.98; 20% remains

Fluorine-18 is a radioactive isotope that decays by positron emission to form oxygen-18 with a half-life of 109.7 min. (A positron is a particle with the mass of an electron and a single unit of positive charge; the equation is ${}_{518}^{\phantom{\rule{1em}{0ex}}9}\text{F}⟶{}_{18}^{\phantom{\rule{0.5em}{0ex}}8}\text{O}+{\text{e}}^{-}.\right)$ Physicians use 18 F to study the brain by injecting a quantity of fluoro-substituted glucose into the blood of a patient. The glucose accumulates in the regions where the brain is active and needs nourishment.

(a) What is the rate constant for the decomposition of fluorine-18?

(b) If a sample of glucose containing radioactive fluorine-18 is injected into the blood, what percent of the radioactivity will remain after 5.59 h?

(c) How long does it take for 99.99% of the 18 F to decay?

Suppose that the half-life of steroids taken by an athlete is 42 days. Assuming that the steroids biodegrade by a first-order process, how long would it take for $\frac{1}{64}$ of the initial dose to remain in the athlete’s body?

252 days

Recently, the skeleton of King Richard III was found under a parking lot in England. If tissue samples from the skeleton contain about 93.79% of the carbon-14 expected in living tissue, what year did King Richard III die? The half-life for carbon-14 is 5730 years.

Nitroglycerine is an extremely sensitive explosive. In a series of carefully controlled experiments, samples of the explosive were heated to 160 °C and their first-order decomposition studied. Determine the average rate constants for each experiment using the following data:

 Initial [C 3 H 5 N 3 O 9 ] ( M ) 4.88 3.52 2.29 1.81 5.33 4.05 2.95 1.72 t (s) 300 300 300 300 180 180 180 180 % Decomposed 52 52.9 53.2 53.9 34.6 35.9 36 35.4

[ A ] 0 ( M ) k $×$ 10 3 (s −1 )
4.88 2.45
3.52 2.51
2.29 2.54
1.81 2.58
5.33 2.35
4.05 2.44
2.95 2.47
1.72 2.43

For the past 10 years, the unsaturated hydrocarbon 1,3-butadiene $\left({\text{CH}}_{\text{2}}=\text{CH}–\text{CH}={\text{CH}}_{2}\right)$ has ranked 38th among the top 50 industrial chemicals. It is used primarily for the manufacture of synthetic rubber. An isomer exists also as cyclobutene:

The isomerization of cyclobutene to butadiene is first-order and the rate constant has been measured as 2.0 $×$ 10 −4 s −1 at 150 °C in a 0.53-L flask. Determine the partial pressure of cyclobutene and its concentration after 30.0 minutes if an isomerization reaction is carried out at 150 °C with an initial pressure of 55 torr.

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